This document describes exact tests for assessing Hardy-Weinberg equilibrium that control type 1 error rates better than chi-squared tests, especially for large sample sizes. It presents efficient computational methods for implementing these exact tests, which sum probabilities of all possible genotype configurations conditional on observed allele counts. These exact tests have been programmed into freely available software for quality control and association testing in large genetic studies.

1. Am. J. Hum. Genet. 76:887–883, 2005
887
Report
A Note on Exact Tests of Hardy-Weinberg Equilibrium
Janis E. Wigginton,1
David J. Cutler,2
and Gonçalo R. Abecasis1
1
Center for Statistical Genetics, Department of Biostatistics, University of Michigan, Ann Arbor; and 2
Institute of Genetic Medicine, Johns
Hopkins University School of Medicine, Baltimore
Deviations from Hardy-Weinberg equilibrium (HWE) can indicate inbreeding, population stratification, and even
problems in genotyping. In samples of affected individuals, these deviations can also provide evidence for association.
Tests of HWE are commonly performed using a simple x2
goodness-of-fit test. We show that this x2
test can have
inflated type I error rates, even in relatively large samples (e.g., samples of 1,000 individuals that include ∼100
copies of the minor allele). On the basis of previous work, we describe exact tests of HWE together with efficient
computational methods for their implementation. Our methods adequately control type I error in large and small
samples and are computationally efficient. They have been implemented in freely available code that will be useful
for quality assessment of genotype data and for the detection of genetic association or population stratification in
very large data sets.
In the absence of migration, mutation, natural selection,
and assortative mating, genotype frequencies at any lo-
cus are a simple function of allele frequencies. This phe-
nomenon, now termed “Hardy-Weinberg equilibrium”
(HWE), was first described in the early part of the twen-
tieth century (Hardy 1908; Weinberg 1908). The orig-
inal descriptions of HWE are an important landmark in
the history of population genetics (Crow 1988), and it
is now common practice to check whether observed
genotypes conform to Hardy-Weinberg expectations.
These expectations appear to hold for most human pop-
ulations, and deviations from HWE at particular mark-
ers may suggest problems with genotyping or population
structure or, in samples of affected individuals, an as-
sociation between the marker and disease susceptibility.
Here, we describe efficient implementations of exact
tests for HWE, which are suitable for use in large-scale
studies of SNP data, even when hundreds of thousands
of markers are examined. The availability of data on
patterns of linkage disequilibrium across the genome
(International HapMap Consortium 2003), interest in
identifying susceptibility alleles for complex diseases
Received November 18, 2004; accepted for publication February
22, 2005; electronically published March 23, 2005.
Address for correspondence and reprints: Dr. Janis E. Wigginton,
Department of Biostatistics, School of Public Health, University of
Michigan, Ann Arbor, MI 48109. E-mail: goncalo@umich.edu
䉷 2005 by The American Society of Human Genetics. All rights reserved.
0002-9297/2005/7605-0017$15.00
(Cardon and Abecasis 2003), and advances in genotyp-
ing technology (Kwok 2001; Weber and Broman 2001)
suggest that such large studies will be increasingly com-
mon. The principles and procedures used for testing
HWE are well established (Levene 1949; Haldane 1954;
Hernandez and Weir 1989; Wellek 2004), but the lack
of a publicly available, efficient, and reliable implemen-
tation for exact tests has led many scientists to rely on
asymptotic tests that can perform poorly with realistic
sample sizes.
Consider a sample of SNP genotypes for N unrelated
diploid individuals measured at an autosomal locus. The
sample includes 2N alleles, including copies of the
nA
rarer allele and copies of the common allele. Let the
nB
number of heterozygous AB genotypes be , and note
nAB
that the numbers of AA and BB homozygous genotypes
are and . Note
n p (n ⫺ n ) / 2 n p (n ⫺ n ) / 2
AA A AB BB B AB
that there are possible arrangements for
(2N)! / n !n !
A B
the alleles in the sample and that nAB
2 N!/(n !n !n !)
AA AB BB
of these arrangements correspond to exactly het-
nAB
erozygotes. Thus, under the assumption of HWE, the
probability of observing exactly heterozygotes in a
nAB
sample of N individuals with minor alleles is
nA
nAB
2 N! n !n !
A B
P(N p n FN, n ) p # . (1)
AB AB A
( )
n !n !n ! 2N !
AA AB BB
This equation holds for each possible number of het-
erozygotes, . When is odd, possible numbers of
n n
AB A

2. 888 Am. J. Hum. Genet. 76:887–883, 2005
heterozygotes are 1, 3, 5,…, . When is even, pos-
n n
A A
sible numbers of heterozygotes are 0, 2, 4,…, . The
nA
expression for given in equation (1) leads
P(n FN,n )
AB A
to natural tests for HWE. For example, one could
define one-sided tests that focus on detection of a de-
ficit of heterozygotes, by calculating the statistic P p
low
, or detection of an excess of heter-
P(N ⭐ n FN,n )
AB AB A
ozygotes, by calculating the statistic P p P(N ⭓
high AB
. In each case, the statistic can be calculated
n FN,n )
AB A
by simply summing over equation (1), to include all pos-
sible values of that are lower (for ) or higher (for
N P
AB low
) than those observed in the actual data. A test for
Phigh
a deficit of heterozygotes in relation to Hardy-Weinberg
expectations is appropriate when deviations from HWE
due to inbreeding or population stratification are sus-
pected, since both of these increase the proportion of
homozygotes in the population. A test for an excess
of heterozygotes is appropriate when one suspects prob-
lems in genotyping due to the existence of highly ho-
mologous regions in the genome, since these low-copy
repeats often lead to an increase in the proportion of
apparent heterozygotes in the sample. In other settings,
it might be appropriate to use both tests. For example,
many technologies score genotypes by clustering signals,
and misspecified clusters can result in either vast excesses
or vast deficits of heterozygotes.
When neither an increase nor a decrease in the pro-
portion of heterozygotes is specifically expected, one
could perform two separate one-sided tests or, instead,
use a two-sided test statistic (Weir 1996). A natural
two-sided test statistic could be defined as P p
2a
. This two-sided statistic is appeal-
min (1.0, 2P , 2P )
high low
ing because it leads to rejection of HWE at significance
level 2a in instances in which the one-sided tests lead to
the rejection of HWE at significance level a. However,
because of the asymmetric nature of the distribution of
heterozygote counts in a sample, the statistic is quite
conservative in practice, and we do not recommend
its use. Instead, an appealing approach, analogous to
Fisher’s exact test for contingency tables (Fisher 1934),
is to calculate the probability of observing a sample con-
figuration that is even less likely than the one being eval-
uated, conditional on the observed allele counts. This
can be achieved using a statistic similar to the Monte
Carlo statistic proposed by Guo and Thompson (1992)
for multiallelic markers:
P p I P(N p n FN,n )
[
冘
HWE AB AB A
∗
nAB
∗
⭓ P(N p n FN,n ) ]
AB AB A
∗
#P(N p n FN,n ) .
AB AB A
In this definition, I[x] is an indicator function that is
equal to 1 when the comparison is true and equal to 0
otherwise. The sum should be performed over all het-
erozygote counts that are compatible with the ob-
∗
nAB
served number of minor alleles, .
nA
Most of the computational effort required for per-
forming exact tests of linkage disequilibrium is spent
evaluating the factorials in equation (1) for each possible
value of . By use of a naive approach, evaluating
nAB
equation (1) requires 5N–6N multiplications and one
division for each possible value of . We simplify cal-
nAB
culations by using the recurrence relationships previ-
ously recognized by Guo and Thompson (1992) in the
implementation of their Markov chain–Monte Carlo
sampler:
P(N p n ⫹ 2FN, n )
AB AB A
4n n
AA BB
p P(N p n FN, n ) , and
AB AB A
(n ⫹ 2)(n ⫹ 1)
AB AB
P(N p n ⫺ 2FN, n )
AB AB A
n (n ⫺ 1)
AB AB
p P(N p n FN, n ) . (2)
AB AB A
4(n ⫹ 1)(n ⫹ 1)
AA BB
In this way, evaluating the probability for each possible
number of heterozygotes takes only four multiplications
and one division, whatever the sample size N. To avoid
underflow, it is best to first calculate the probability of
observing the expected number of heterozygotes (in this
case, the most likely outcome) and then use the recur-
rence relationships to calculate probabilities for all other
outcomes. A further reduction of computational effort
is possible by noting that one need only calculate relative
probabilities for each outcome and then scale these to
ensure that their sum is 1.0. This means that the prob-
ability of observing the expected number of heterozy-
gotes can be replaced with an arbitrary constant when
using the recurrence relations in equation (2), provided
that the final result is scaled.
Table 1 illustrates the performance of the statistics for
a sample of 100 individuals in which 21 copies of the
minor allele are present. The observed number of het-
erozygotes will vary from 1 to 21 and must be odd. Note
that only a small number of distinct sample configura-
tions are possible, and each of these is associated with
a specific probability for the exact tests. If the desired
significance level a does not correspond exactly to one
of these discrete outcomes, then the exact test statistics
will be conservative (Hernandez and Weir 1989). For
example, at the significance level , the PHWE and
a p 0.05
Plow statistics both reject the hypothesis of HWE if ⭐13
heterozygotes are observed in this setting. Since the prob-
ability of observing ⭐13 heterozygotes is 0.010, the tests
are conservative. In contrast, the asymptotic x2
test sta-
tistic results in rejection of HWE when ⭐15 heterozy-

3. Figure 1 Type I error rates as a function of minor-allele counts for rare alleles, for samples of either 100 or 1,000 chromosomes and corresponding to a significance threshold of
, 0.01, or 0.001. Results are plotted as a function of the number of minor alleles in the sample for the exact statistic (red) and for the asymptotic x2
test statistic (blue). A gray
a p 0.05 PHWE
line denotes the nominal error rate. Note that the Y-axes in figures 1 and 2 differ.

4. 890 Am. J. Hum. Genet. 76:887–883, 2005
Table 1
Possible Sample Configurations and Their Probabilities for a Sample of 100
Individuals and 21 Minor-Allele Copies Are Tabulated
NO. OF
HETEROZYGOTES
(nAB) PROBABILITY
a
x2
TEST P
EXACT TEST P VALUES
PHWE Phigh Plow
5 !.000001 !.000001b
!.000001b
1.000000 !.000001b
7 .000001 !.000001b
.000001b
1.000000 .000001b
9 .000047 !.000001b
.000048b
.999999 .000048b
11 .000870 .000039b
.000919b
.999952 .000919b
13 .009375 .002228b
.010293b
.999081 .010293b
15 .059283 .045180b
.069576 .989707 .069576
17 .214465 .342972 .284042 .930424 .284042
19 .406355 .906529 1.000000 .715958 .690396
21 .309604 .244336 .593645 .309604 1.000000
NOTE.—The probability of observing each possible outcome is given, together with
the corresponding P values for tests of HWE based on the x2
statistic and on the exact
test statistics PHWE, Plow, and Phigh (described in the main text).
a
.
P(n FN p 100,n p 21)
AB A
b
Configurations that would be rejected at the significance level a p 0.05.
gotes are observed (for ⭐15 heterozygotes, the x2
test
statistic corresponds to an asymptotic ). This
P ⭐ .045
results in an inflated type I error rate of 0.070 and there-
fore is inappropriate. In this sample, it is not possible
to reject HWE because of an excess of heterozygous
individuals—the probability of observing the maximum
of 21 heterozygotes is 0.31, and none of the test statistics
gives a P value !.05 for this extreme configuration. Ad-
ditional examples of the performance of exact test sta-
tistics for HWE can be found in the work by Vithayasai
(1973).
In general, the exact test statistics are conservative
when a small number of minor-allele copies are present
in the sample, but they approximate nominal signifi-
cance levels as the sample size (and number of minor-
allele copies) increases. In contrast, the commonly used
x2
statistic can produce excessively small or large P val-
ues for specific outcomes (Hernandez and Weir 1989).
To comprehensively evaluate the performance of the x2
and exact test statistics, we calculated their type I error
rates for specified significance levels of , 0.01,
a p 0.05
or 0.001, for sample sizes of or
N p 100 N p 1,000
individuals and varying minor-allele counts. The results
are summarized in figure 1 (for samples in which !25%
of chromosomes carry the minor allele) and figure 2 (for
samples in which 110% of chromosomes carry the mi-
nor allele), and it is clear that the statistics exhibit some
periodicity in their type I error rates. As expected, both
the exact PHWE statistic and the x2
statistic perform better
as the sample size and minor-allele counts increase. Nev-
ertheless, one important difference is that the x2
statistic
can sometimes be extremely anticonservative (e.g., in a
sample of 1,000 individuals, when nominal ,
a p 0.001
the true type I error rate can exceed 0.06 and is often
10.01 for minor-allele counts !100), whereas the exact
statistic never exceeds the nominal significance level. In
practical settings, the x2
statistic could lead to many false
rejections of HWE that depend on only the particular
count of minor alleles in the sample.
To understand the periodicity of the statistics, it is
important to consider the discrete nature of the data.
For example, for a sample of individuals in-
N p 100
cluding 2–5 copies of the minor allele, we reject HWE
at the significance level (fig. 1A) when there
a p 0.05
is at least one homozygote for the minor allele. The
probability of observing more than one homozygote for
the minor allele increases gradually from 0.0050 when
there are two copies of the allele in the sample up to
0.0499 when there are five copies of the minor allele in
the sample. When there are 6–14 copies of the minor
allele in the sample, we reject HWE at the a p 0.05
significance level (fig. 1A) when at least two homo-
zygotes for the rare allele are observed. Again, the prob-
ability of a more extreme event is quite low for small
numbers of the rare allele ( with six copies of
P p .0011
the minor allele in the sample) but gradually increases
if there are additional copies of the minor allele in the
sample ( with 13 copies of the minor allele).
P p .0482
In table 2, the overall type I error rates for each sta-
tistic are summarized for sample sizes of 100 or 1,000
individuals and various ranges of minor-allele counts. It
is clear that, on average, the x2
test approximates nom-
inal significance levels as the number of minor alleles in
the sample increases. Nevertheless, as illustrated in figure
1, this is achieved at the cost of inflated error rates for
samples with specific numbers of minor alleles. Even in
a sample of 1,000 individuals, the type I error rate at a
p 0.001 for the x2
test is inflated when there are !200
copies of the minor allele (corresponding to an allele
frequency of ∼10%). The exact tests approximate nom-

5. Figure 2 Type I error rates as a function of minor-allele counts for common alleles, for samples of either 100 or 1,000 chromosomes and corresponding to a significance threshold of
, 0.01, or 0.001. Results are plotted as a function of the number of minor alleles in the sample for the exact statistic (red) and for the asymptotic x2
test statistic (blue). A gray
a p 0.05 PHWE
line denotes the nominal error rate. Note that the Y-axes in figures 1 and 2 differ.

6. 892 Am. J. Hum. Genet. 76:887–883, 2005
Table 2
Actual Error Rates for the x2
Test Statistic and the PHWE Test Statistic for Nominal
Significance Level a p 0.01 or 0.001
SAMPLE AND
MINOR-ALLELE COUNT
a p 0.01a
a p 0.001a
x2
PHWE x2
PHWE
N p 1,000
1–100 .0208b
(.0208)b
.0039 (.0039) .0088b
(.0088)b
.0004 (.0004)
101–200 .0100 (.0154)b
.0065 (.0052) .0017b
(.0053)b
.0006 (.0005)
201–400 .0097 (.0126)b
.0083 (.0067) .0010 (.0032)b
.0008 (.0006)
401–1,000 .0100 (.0110)b
.0090 (.0081) .0010 (.0018)b
.0009 (.0008)
N p 100
1–10 .0292b
(.0292)b
.0024 (.0024) .0114b
(.0114)b
.0001 (.0001)
11–20 .0191b
(.0242)b
.0035 (.0030) .0035b
(.0074)b
.0003 (.0002)
21–40 .0083 (.0162)b
.0037 (.0033) .0016b
(.0045)b
.0004 (.0003)
41–100 .0099 (.0124)b
.0072 (.0057) .0009 (.0023)b
.0006 (.0005)
NOTE.—Results are tabulated for samples of 100 and 1,000 individuals and represent simple
averages for each range of minor-allele counts.
a
The error rate for each bin is tabulated, followed by the cumulative error rate in parenthesis.
The cumulative error rate is calculated by including each bin and all previous bins. For example,
for a sample of size 1,000, when a p 0.001, the type I error rate for the standard x2
test in a
sample with 101–200 copies of the minor allele is 0.0017 and the cumulative error rate, cor-
responding to samples with 1–200 copies of the minor allele, is 0.0053.
b
Exceeds nominal significance level.
inal significance levels with increasing sample size but
remain conservative because of the discrete nature of the
data.
As a final evaluation of our approach, we applied our
method to a subset of the genotypes collected by the
International HapMap Consortium (2003). We focused
on a set of 18,460 SNP markers genotyped indepen-
dently by two different centers with no discrepancies
between the two sets of experimental results. For each
of these markers, we evaluated evidence against HWE
by using both the exact PHWE statistic and the asymptotic
x2
statistic. Results were broadly similar for 14,889
markers with minor-allele frequencies ⭓20%. However,
we observed noticeable differences for 3,571 markers
with minor-allele frequencies !20%. For example, the
x2
test rejected HWE for 71 of these markers at a p
(twice as many as the 35 markers expected to fail
0.01
this test by chance), whereas the exact test rejected HWE
for only 33 markers. At the more stringent a p 0.001
significance level, the x2
test rejected HWE for 28 mark-
ers (rejection for 3 markers is expected by chance),
whereas the exact PHWE statistic rejected HWE for only
5 markers.
Although we focus on testing the agreement of ob-
served genotypes with HWE proportions, computation-
ally efficient exact tests can be constructed for any de-
sired genotype proportions. In brief, let the expected
proportion of heterozygotes be pAB and the two ho-
mozygote proportions be pAA and pBB. For exam-
ple, in a population with inbreeding coefficient f, we
might expect the proportion of heterozygotes to be
. Define the quantity so that
2
2(1 ⫺ f)p p v p p / p p
A B AB AA BB
when HWE holds. Then, the probability of ob-
v p 4
serving nAB heterozygotes is
n /2
AB
v N! 1
P(N p n FN,n ) p # ,
AB AB A
n !n !n ! C
AA AB BB
where
∗
n /2
AB
v N!
C p 冘 ∗ ∗ ∗
∗ n !n !n !
n AA AB BB
AB
(Wellek 2004). It is simple to verify that the recurrence
relationships given in equation (2) can be extended to
this setting by replacing the number 4 with the quantity
v in each expression.
The exact test statistics for HWE described here are
accurate for a variety of allele frequencies and can be
computed in an inexpensive manner. We recommend
that they be used instead of the standard x2
test statistic
in all situations. For large data sets, rather than fixing
an arbitrary threshold for rejecting HWE, we suggest
that methods based on the false-discovery rate (Benja-
mini and Hochberg 1995) be used to identify a subset
of markers whose genotypes do not conform to the ex-
pected equilibrium distribution.
The PHWE test statistic described here is implemented
in the Pedstats software package (see Pedstats Web site),
which generates summaries and checks the integrity of
genetic data. In addition, code for calculating Plow, Phigh,
and PHWE in C/C⫹⫹, R, and Fortran is available from
the authors’ Web site. With appropriate citation, our
code is freely available for use and can be incorporated

7. Reports 893
into other programs. The HapMap Project genotype
data are freely available at the HapMap Web site.
Acknowledgments
We gratefully acknowledge grant support from the National
Human Genome Research Institute and the National Eye In-
stitute. The manuscript was improved by helpful comments
from reviewers.
Electronic-Database Information
The URLs for data presented herein are as follows:
Authors’ Web site, http://www.sph.umich.edu/csg/abecasis/
HapMap, http://www.hapmap.org/
Pedstats, http://www.sph.umich.edu/csg/abecasis/Pedstats/
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